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Simplifying 2x2 + 23 = 14x Reorder the terms: 23 + 2x2 = 14x Solving 23 + 2x2 = 14x Solving for variable 'x'. Reorder the terms: 23 + -14x + 2x2 = 14x + -14x Combine like terms: 14x + -14x = 0 23 + -14x + 2x2 = 0 Begin completing the square. Divide all terms by 2 the coefficient of the squared term: Divide each side by '2'. 11.5 + -7x + x2 = 0 Move the constant term to the right: Add '-11.5' to each side of the equation. 11.5 + -7x + -11.5 + x2 = 0 + -11.5 Reorder the terms: 11.5 + -11.5 + -7x + x2 = 0 + -11.5 Combine like terms: 11.5 + -11.5 = 0.0 0.0 + -7x + x2 = 0 + -11.5 -7x + x2 = 0 + -11.5 Combine like terms: 0 + -11.5 = -11.5 -7x + x2 = -11.5 The x term is -7x. Take half its coefficient (-3.5). Square it (12.25) and add it to both sides. Add '12.25' to each side of the equation. -7x + 12.25 + x2 = -11.5 + 12.25 Reorder the terms: 12.25 + -7x + x2 = -11.5 + 12.25 Combine like terms: -11.5 + 12.25 = 0.75 12.25 + -7x + x2 = 0.75 Factor a perfect square on the left side: (x + -3.5)(x + -3.5) = 0.75 Calculate the square root of the right side: 0.866025404 Break this problem into two subproblems by setting (x + -3.5) equal to 0.866025404 and -0.866025404.Subproblem 1
x + -3.5 = 0.866025404 Simplifying x + -3.5 = 0.866025404 Reorder the terms: -3.5 + x = 0.866025404 Solving -3.5 + x = 0.866025404 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '3.5' to each side of the equation. -3.5 + 3.5 + x = 0.866025404 + 3.5 Combine like terms: -3.5 + 3.5 = 0.0 0.0 + x = 0.866025404 + 3.5 x = 0.866025404 + 3.5 Combine like terms: 0.866025404 + 3.5 = 4.366025404 x = 4.366025404 Simplifying x = 4.366025404Subproblem 2
x + -3.5 = -0.866025404 Simplifying x + -3.5 = -0.866025404 Reorder the terms: -3.5 + x = -0.866025404 Solving -3.5 + x = -0.866025404 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '3.5' to each side of the equation. -3.5 + 3.5 + x = -0.866025404 + 3.5 Combine like terms: -3.5 + 3.5 = 0.0 0.0 + x = -0.866025404 + 3.5 x = -0.866025404 + 3.5 Combine like terms: -0.866025404 + 3.5 = 2.633974596 x = 2.633974596 Simplifying x = 2.633974596Solution
The solution to the problem is based on the solutions from the subproblems. x = {4.366025404, 2.633974596}
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